The length of the subnormal to any point on a curve is always constant. Then,the curve is . . . . . . .

The length of the subtangent at any point of the curve $x^my^n = a^{m+n}$ is proportional to

The coordinates of a point on the curve $y = x \log x$ at which the normal is parallel to the line $2x - 2y = 3$ are

Find the slope of the tangent to the curve $y = x^{3} - x + 1$ at the point where the $x$-coordinate is $2$.

The abscissa of the point on the curve $\sqrt{xy} = a + x$ where the tangent cuts off equal intercepts from the coordinate axes is $(a > 0)$.

If all the normals drawn to the curve $y=\frac{1+3x^2}{3+x^2}$ at the points of intersection of $y=\frac{1+3x^2}{3+x^2}$ and $y=1$ pass through the point $(\alpha, \beta)$,then $3\alpha+2\beta=$